Minitest 3 question 2c

Physics Q&A
1

Dear courseteam,

I am still a bit confused about the answers of minitest 3 2c.
From question 2b we know that the dispersion relation is E = ϵ0 − 2t1(cos(kxa) + cos(kya)) − 4t2(cos(kxa) cos(kya)).
I approached the question as the following: the two cosines form a bandstructure by sometimes overlapping and sometimes not (due to a possible different amplitude if t2 uneq to t1)
I have two questions about this:

  1. I don’t understand how we can see this as 1 (classical) band since the 2 cosinus would not always be overlapping
  2. even if we would say that the cosinus always overlap wouldn’t we then need 4 electrons to completely fill this band? Therefore if the material was an insulator it would need 0 or 4 electrons right? (compared to the answer of 0 or 2 electrons)
2

Hi,
The single band you obtain form that dispersion relation is given by substituting at each (kx, ky) point and obtaining an energy. You can try and plot this for different values of t1 and t2.
You are trying to see whether the cosines add up or not as if they could go different ways. But the band is given by E(kx, ky) which has a single value.

Does that clarify your doubt?

3

Ahh of course yes I understand it now, thank you!